On the perturbation of spectra
On the perturbation of spectra
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关于光谱的扰动
DOI：
10.1002/cpa.3160180402
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发表时间：
1965
影响因子：
3
通讯作者：
W. Donoghue
中科院分区：
文献类型：
--
作者：
W. Donoghue
本文件的一个重要部分是审查的结果所获得的各种作者有关的变化时发生的自伴算子在希尔伯特空间受到扰动的某些类型。几乎所有的定理，我们提到的是在相对特殊的情况下获得的，他们的证明是相应的困难。本论文的机器使这样的定理在一次更一般和更透明。我们发现，当需要比较给定算子族中算子的谱时，三种明显不同的情形是完全等价的，即1）由自伴算子H通过增加一维值域Ht= Ht + tP的投影的倍数而得到的算子族，2）由具有亏指数（1，1）的对称算子的所有自伴扩张而得到的算子族，3）与给定的Sturm-Liouville问题的极限点相关联的自伴扩张族。我们的机器使我们能够给出最一般的模型的对称算子的亏损指数（1，1）和家庭的自伴扩张。特别地，奇异Sturm-Liouville算子的Weyl理论中的极限点解析函数m（z）可以推广到所有亏指数为（1，1）的对称算子.文末的参考文献构成了对文献的充分介绍。作者还受益于与N. Aronszajn，M. Schreiber和P. Rejto。
A substantial part of the present paper is a review of results obtained by a variety of authors concerning the changes which occur when a selfadjoint operator in Hilbert space is subjected to perturbations of certain types. Almost all of the theorems to which we refer were obtained in relatively special situations and their proofs were correspondingly difficult. The machinery of the present paper makes such theorems at once more general and more transparent. We find that three apparently different situations are quite equivalent when it is required to compare the spectra of operators in a given family, namely 1) the family of operators obtained from a selfadjoint H, by adding a multiple of a projection of onedimensional range Ht= H,,+ tP, 2) the family of operators obtained by taking all selfadjoint extensions of a symmetric operator with deficiency indices (1, 1), and 3) the family of selfadjoint extensions associated with a given limit-point Sturm-Liouville problem. Our machinery enables us to give the most general model of a symmetric operator with deficiency indices (1, 1) and its family of selfadjoint extensions. In particular, the analytic function m (z) which is the limit point in the Weyl theory of singular Sturm-Liouville operators admits an obvious generalization to all symmetric operators with deficiency indices (1, 1). The references at the end of the paper form an adequate introduction to the literature. The author has also benefited from convemafiohs with N. Aronszajn, M. Schreiber and P. Rejto.